Overview
This lecture covers techniques for evaluating inverse trigonometric expressions by constructing triangles and applying relationships between trigonometric and inverse trigonometric functions.
Evaluating Inverse Trig Functions Geometrically
- Inverse sine (arcsin) expressions can be interpreted by constructing a right triangle with known side ratios.
- The sine of an angle equals the ratio of the opposite side to the hypotenuse in a right triangle.
- Example: arcsin(√3/2) can be visualized as a 60° (π/3 radians) angle in a 1-√3-2 triangle.
- Inverse cosine (arccos) with negative values can also be approached using triangle construction and angle properties.
- Example: arccos(-√3/2) yields 150° (5π/6 radians).
Handling Less Familiar Inverse Trig Functions
- Arcsecant (arcsec) is evaluated by recognizing that secant is the reciprocal of cosine.
- Example: arcsec(2) becomes arccos(1/2), giving π/3 radians as the answer.
Composition of Trig and Inverse Trig Functions
- For compositions like arcsin(sin(8π/3)), the output must fall within the restricted range of arcsin (−π/2 to π/2).
- Use the unit circle to find the equivalent angle within this range, e.g., arcsin(sin(8π/3)) = π/3.
- Graphical methods confirm the same result using the periodicity and symmetry of sine.
General Triangle Constructions for Function Compositions
- For cos(arcsin(r)), visualize a right triangle with opposite = r and hypotenuse = 1; adjacent = √(1−r²).
- Result: cos(arcsin(r)) = √(1−r²).
- For cos(arctan(x)), use a right triangle with opposite = x and adjacent = 1; hypotenuse = √(1+x²).
- Result: cos(arctan(x)) = 1/√(1+x²).
Key Terms & Definitions
- Arcsin (Inverse Sine) — returns the angle whose sine is a given number; range: [−π/2, π/2].
- Arccos (Inverse Cosine) — returns the angle whose cosine is a given number; range: [0, π].
- Arcsec (Inverse Secant) — returns the angle whose secant is a given number.
- Unit Circle — a circle with radius 1 used to understand trigonometric functions.
- Pythagoras' Theorem — a² + b² = c² for right triangles.
Action Items / Next Steps
- Practice sketching triangles for various inverse trig expressions.
- Review how to convert less familiar trig functions (like secant) to sines and cosines.
- Ensure understanding of trig function ranges and periodicity when composing functions.